Linear stochastic differential equations in the dual of a multi-Hilbertian space
We prove the existence and uniqueness of strong solutions for linear stochastic differential equations in the space dual to a multi–Hilbertian space driven by a finite dimensional Brownian motion under relaxed assumptions on the coefficients. As an application, we consider equtions in S' with c...
Saved in:
| Date: | 2008 |
|---|---|
| Main Authors: | , , |
| Format: | Article |
| Language: | English |
| Published: |
Інститут математики НАН України
2008
|
| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/4549 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Linear stochastic differential equations in the dual of a multi-Hilbertian space / L. Gawarecki, V. Mandrekar, B. Rajeev // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 2. — С. 28–34. — Бібліогр.: 9 назв.— англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859896660767801344 |
|---|---|
| author | Gawarecki, L. Mandrekar, V. Rajeev, B. |
| author_facet | Gawarecki, L. Mandrekar, V. Rajeev, B. |
| citation_txt | Linear stochastic differential equations in the dual of a multi-Hilbertian space / L. Gawarecki, V. Mandrekar, B. Rajeev // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 2. — С. 28–34. — Бібліогр.: 9 назв.— англ. |
| collection | DSpace DC |
| description | We prove the existence and uniqueness of strong solutions for linear stochastic differential equations in the space dual to a multi–Hilbertian space driven by a finite dimensional Brownian motion under relaxed assumptions on the coefficients. As an application, we consider equtions in S' with coefficients which are differential operators violating the typical growth and monotonicity conditions.
|
| first_indexed | 2025-12-07T15:55:03Z |
| format | Article |
| fulltext |
Theory of Stochastic Processes
Vol. 14 (30), no. 2, 2008, pp. 28–34
UDC 519.21
L. GAWARECKI, V. MANDREKAR, AND B. RAJEEV
LINEAR STOCHASTIC DIFFERENTIAL EQUATIONS
IN THE DUAL OF A MULTI-HILBERTIAN SPACE
We prove the existence and uniqueness of strong solutions for linear stochastic dif-
ferential equations in the space dual to a multi–Hilbertian space driven by a finite
dimensional Brownian motion under relaxed assumptions on the coefficients. As an
application, we consider equtions in S′
with coefficients which are differential oper-
ators violating the typical growth and monotonicity conditions.
1. Assumptions
We consider a countably Hilbertian space (Φ, τ), whose topology τ is determined by a
family of separable Hilbertian seminorms ‖ · ‖p, p ∈ R (for a detailed exposition, see [4]).
For any p ∈ R+, we identify φ ∈ Φ with [φ]p ∈ Φ/ ker ‖ · ‖p and denote the completion
of Φ in ‖ · ‖p by Hp. Then Hp is a real separable Hilbert space containing Φ as its dense
subspace, and the embedding (Φ, τ) ↪→ (Hp, ‖ ·‖p) is continuous. Assume that, for q ≤ p,
the canonical embedding (Hp, ‖ · ‖p) ↪→ (Hq, ‖ · ‖q) is continuous, i.e., ‖ · ‖p dominates
‖ · ‖q, denoted by ‖ · ‖q ≺ ‖ · ‖p.
In applications, the strong dual Φ
′
of Φ is realized through Hilbert spaces H−p iso-
morphic to H
′
p, as Φ
′
=
⋃
p∈R+
H−p, where
Φ ⊂ Hp ⊂ H0 ⊂ H−p ⊂ Φ
′
,
and all the inclusions are continuous. The Hilbert spaces Hp and H−p are dual, in the
pairing
Hp〈hp, h−p〉H−p , hp ∈ Hp, h
−p ∈ H−p,
being an extension of the duality between Φ and Φ
′
.
Assume there exists a total set {φj}∞j=1 in Φ, which is a common orthogonal system
for all Hilbert spaces Hp, p ∈ R, and denote, by {hp
j} = ‖φj‖−1
p φj , the ONB in Hp
derived from φj . We set Φ〈φn, φn〉Φ′ = ‖φn‖2
0 = 1. For f ∈ Φ, the scalar product in Hp,
p ∈ R, can be calculated as 〈f, hp
n〉p = 〈f, φn〉0‖φn‖p.
For linear topological vector spaces A and B, we denote, by L(A,B), the space of
continuous linear operators from A to B. For a bounded linear operator T ∈ L
(
Rd, Hp
)
,
its Hilbert–Schmidt norm is calculated as ‖T ‖HS(p) =
(∑d
i=1 ‖Tei‖2
p
)1/2, where {ei}d
i=1
is the canonical basis in Rd.
We will study a stochastic process with values in Φ and Φ
′
. Let (Ω,F , {Ft}t≥0, P ) be
a filtered probability space satisfying the usual conditions: F0 contains all A ∈ F , such
that P (A) = 0, and Ft =
⋂
s>t Fs. Measurability will be understood with respect to
the Borel σ–fields BΦ, BΦ′ (respectively) and this filtered probability space. Since Φ is
2000 AMS Mathematics Subject Classification. Primary 60H15.
Key words and phrases. Infinite dimensional stochastic differential equations, multi-Hilbertian spa-
ces, existence, uniqueness, monotonicity.
28
LINEAR SDE’S IN THE DUAL OF A MULTI-HILBERTIAN SPACE 29
a countable multi–Hilbertian space, the Borel σ–fields on Φ
′
generated by strongly open
sets and by weakly open sets coincide.
For 0 ≤ t ≤ T , consider the functions
L : [0, T ] × Ω → L(Φ′,Φ′), A : [0, T ] × Ω → L
(
Φ′, L(Rd,Φ′)
)
We introduce the following conditions on L and A. Below, let q ≤ p.
1. (Invariance [INV(Φ)]) Φ is invariant for L and A, i.e. L(t, ω) : Φ → Φ and
A(t, ω) : Φ → L(Rd,Φ).
2. (Measurability [MR(Φ
′
)]) For any progressively measurable Φ–valued process
{Xt}t≤T and any x ∈ Rd, {L(t, ω)Xt(ω)}t≤T and {A(t, ω)Xt(ω)x}t≤T are Φ
′
–
valued progressively measurable processes.
3. (Measurability [MR(p,q)]) For any progressively measurable Hp–valued process
{Xt}t≤T and any x ∈ Rd, {L(t, ω)Xt(ω)}t≤T and {A(t, ω)Xt(ω)x}t≤T are Hq–
valued progressively measurable processes.
4. (Boundedness [B(p,q)]) L : [0, T ] × Ω → L(Hp, Hq) and A : [0, T ] × Ω →
L(Hp, L(Rd, Hq)) and L and A are uniformly bounded, i.e.
‖L(t, ω)u‖q + ‖A(t, ω)u‖HS(q) ≤ θ‖u‖p
∀u ∈ Hp , 0 ≤ t ≤ T and ω ∈ Ω, with θ depending only on p and q.
5. (Monotonicity [M(p)])
2〈u, L(t, ω)u〉p + ‖A(t, ω)u‖2
HS(p) ≤ θ‖u‖2
p
∀u ∈ Φ , 0 ≤ t ≤ T and ω ∈ Ω, with θ depending only on p.
6. (Monotonicity [M(p,q)]) L : [0, T ] × Ω → L(Hp, Hq) and A : [0, T ] × Ω →
L(Hp, L(Rd, Hq)), and
2〈u, L(t, ω)u〉q + ‖A(t, ω)u‖2
HS(q) ≤ θ‖u‖2
q
∀u ∈ Hp, 0 ≤ t ≤ T and ω ∈ Ω, with θ depending only on p and q.
Condition [B(p,q)] is very weak, since the growth of A(t, ω) in Hq is bounded by the
norm of the argument in Hp, and ‖ · ‖p � ‖ · ‖q. This weakness in the growth condition
is the major difficulty in proving the existence result. Note, for example, that one part
of the linear growth condition in Kallianpur et al. [5] is stated within the same space.
However, operators as basic as differentiation in S ′
fail to satisfy such growth condition.
2. Existence and Uniqueness of the Solution
Let {Bt, t ≥ 0} be a given d-dimensional standard Brownian motion with respect
to {Ft}t≥0. Let H be a Hilbert space. We denote, by
∫ t
0 Ψ(s) dBs, the stochastic
integral of an L(Rd, H)–valued process Ψ(t), w.r.t. Bt. Note that
∫ t
0 Ψ(s) dBs =∑d
i=1
∫ t
0
Ψ(s)eidB
i
s, where ei is the standard ONB in Rd. The integrals on the RHS
are the integrals of the H–valued processes Ψ(t)ei with respect to the real-valued pro-
cesses Bi
t.
We consider the following stochastic differential equation in Φ
′
:
(2.1)
{
dXt = L(t)Xtdt+A(t)XtdBt
X0 = φ.
The initial condition φ is a Φ
′
–valued F0–measurable random variable.
Definition 1. Let q ≤ p ∈ R and φ(ω) ∈ Hp for all ω ∈ Ω. Assume that the co-
efficients of Eq. (2.1) satisfy conditions [MR(p,q)] and [B(p,q)]. An Hp-valued Ft–
progressively measurable stochastic process {Xt}0≤t≤T defined on a filtered probability
30 L. GAWARECKI, V. MANDREKAR, AND B. RAJEEV
space (Ω,F , {Ft}t≤T , P ) is a strong solution of Eq. (2.1) in Hq if E
∫ T
0 ‖Xt‖2
p dt < ∞
and the following equation holds in Hq:
(2.2) Xt = φ+
∫ t
0
L(s)Xsds+
∫ t
0
A(s)XsdBs for almost all (t, ω).
Conditions [MR(p,q)], [B(p,q)], and progressive measurability assumed in Definition 1
guarantee that the integrals in Eq. (2.2) are well-defined Ft-adapted continuous Hq-
valued processes. Thus, the strong solution has a continuous version in Hq (and, hence,
a progressively measurable version in Hq).
We use techniques similar to those found in [6], [7], and [9]. The next lemma discusses
properties of a solution to an SDE, whose coefficients satisfy the monotonicity condition.
Lemma 1. (Part 1) Assume that the coefficients L and A of Eq. (2.1) satisfy conditions
[INV(Φ)], [MR(Φ
′
)], [M(r)]. Let φ(ω) ∈ Φ for all ω and E‖φ‖2
r < ∞. If {Xt} is a Φ–
valued process satisfying Eq. (2.2) in Hr, for each t ≥ 0, a.s., in the usual sense of an SDE
in a Hilbert space (in particular Xt is continuous in Hr, P (
∫ T
0 ‖L(s)Xs‖r ds <∞) = 1,
and P (
∫ T
0 ‖A(s)Xs‖2
HS(r) ds <∞) = 1), then
(2.3) sup
t≤T
E‖Xt‖2
r ≤ CE‖φ‖2
r.
(Part 2) Let r ≥ p ≥ q. Assume that the coefficients L and A of Eq. (2.1) satisfy
conditions [MR(r,p)], [M(r,p)], [M(p,q)], [B(p,q)], and that E‖φ‖2
p <∞. Let {Xt}0≤t≤T
be an Hr–valued process satisfying Eq. (2.1) in Hp. Let {Yt}0≤t≤T be the continuous
version of {Xt}0≤t≤T in Hp defined by the RHS of (2.2). Then
(2.4) E sup
t≤T
‖Yt‖2
q ≤ CE‖φ‖2
p.
Proof. (Part 1) Using Itô’s formula for ‖ · ‖2
r and condition [M(r)], we obtain
(2.5) ‖Xt‖2
r ≤ ‖φ‖2
r +
∫ t
0
θ‖Xs‖2
r ds+ 2
∫ t
0
d∑
j=1
〈
Xs, A(s)Xs(ej)
〉
r
dBj
s .
Let {τn}∞n=1 be stopping times localizing the local martingale represented by the sto-
chastic integral above, then
E‖Xt∧τn‖2
r ≤ E‖φ‖2
r +
∫ t
0
Eθ‖Xs∧τn‖2
r ds.
Using Gronwall’s lemma and the fact that τn → ∞, we obtain (2.3).
(Part 2) By repeating the proof of (2.3) with the condition [M(r,p)] replacing [M(r)], we
arrive at
sup
t≤T
E‖Yt‖2
p ≤ CE‖φ‖2
p
for the Hp-continuous version Yt of the Hr–valued solution Xt. Since Hp ↪→ Hq, and
‖ · ‖q ≺ ‖ · ‖p, Yt is an Hp–valued process satisfying Eq. (2.2) in Hq. Thus, in (2.5),
we can replace the r–norm with the q–norm, by using condition [M(p,q)]. Consider the
stochastic integral in (2.5). It follows from Burkholder’s inequality, assumption [B(p,q)],
and the bound for E‖Yt‖2
p that
E sup
t≤T
∣∣∣∫ t∧τn
0
d∑
j=1
〈
Ys, A(s)Ys(ej)
〉
q
dBj
s
∣∣∣
≤ CE
(∫ T
0
( d∑
j=1
‖Ys∧τn‖q‖A(s ∧ τn)Ys∧τn(ej)‖q
)2
ds
) 1
2
LINEAR SDE’S IN THE DUAL OF A MULTI-HILBERTIAN SPACE 31
≤ CE
((
sup
t≤T
‖Yt∧τn‖2
q
) 1
2
(∫ T
0
‖Ys‖2
p ds
) 1
2
)
≤ C
2
(
εE sup
t≤T
‖Yt∧τn‖2
q +
1
ε
E
∫ T
0
‖Ys‖2
p ds
)
≤ C
2
(
εE sup
t≤T
‖Yt∧τn‖2
q +
1
ε
E‖φ‖2
p
)
for any ε > 0. Because ‖ · ‖q ≺ ‖ · ‖p, we have
E sup
t≤T
‖Yt∧τn‖2
q ≤ E‖φ‖2
q + E
∫ T
0
θ‖Yt∧τn‖2
q ds+
C
2
(
εE sup
t≤T
‖Yt∧τn‖2
q +
1
ε
E‖φ‖2
p
)
≤ CE‖φ‖2
p +
1
2
E sup
t≤T
‖Yt∧τn‖2
q,
since ε > 0 is arbitrary. The constant C depends only on q, p, and T and can change its
value from line to line. Thus
E sup
t≤T
‖Yt∧τn‖2
q ≤ CE‖φ‖2
p,
and (2.4) follows by Fatou’s lemma.
We will use the same symbol Xt to denote the Hr–valued solution satisfying (2.1) in
Hp and its Hp-continuous version. We now state our main result.
Theorem 1. Let the coefficients A and L of Eq. (2.1) satisfy conditions [INV(Φ)],
[MR(Φ
′
)], [MR(r,p)], [B(r,p)], and [M(r)], for some r ≥ p. Assume that E‖φ‖2
r <
∞. Then equation (2.1) has an Hr–valued strong solution Xt in Hp. If in the above
assumptions [M(p)] holds instead of [M(r)], then the solution is unique.
If, in addition, there exists q ≤ p, such that A and L satisfy conditions [M(p,q)] and
[B(p,q)], then Xt viewed as a continuous Hp–valued strong solution of Eq. (2.1) satisfying
Eq. (2.2) in Hq, is continuous with respect to the initial condition, i.e. for the initial
conditions φn → φ in L2(Ω, Hp), the corresponding solutions Xn(t) and Xt satisfy
Xn → X in L2(Ω, C([0, T ], Hq)).
Proof. Uniqueness follows from the argument provided in Krylov and Rozovskii [6].
Let p ≤ r and X1
t , X2
t ∈ C ([0, T ], Hp) be (continuous versions of) two Hr–valued
strong solutions of Eq. (2.2) in Hp. We denote Yt = X1
t −X2
t and apply Itô’s formula to
‖Yt‖2
p, to obtain
‖Yt‖2
p =
∫ t
0
{
2〈L(s)Ys, Ys〉p + ‖A(s)Ys‖2
HS(p)
}
ds+Mt,
where Mt is a local L2–martingale. We apply Itô’s formula again and obtain
e−μt‖Yt‖2
p = −μ
∫ t
0
‖Ys‖2
pe
−μs ds+
∫ t
0
{
2〈L(s)Ys, Ys〉p + ‖A(s)Ys‖2
HS(p)
}
e−μs ds
+
∫ t
0
e−μs dMs.
Since conditions [M(p)] and [B(r,p)] imply [M(r,p)], taking μ > θ in the latter condition
gives
e−μt‖Yt‖2
p ≤
∫ t
0
e−μs dMs.
Using Doob’s inequality for the non–negative continuous local martingale
Nt =
∫ t
0
e−μs dMs,
32 L. GAWARECKI, V. MANDREKAR, AND B. RAJEEV
we have sup0≤t≤T {Nt} = 0, P–a.s., and the pathwise uniqueness follows.
To prove the existence, we let Pn to be an orthogonal projection of Hp on an n–
dimensional subspace of Φ, spanned by {hp
1, . . . , h
p
n}, Pnu =
∑n
k=1〈u, h
p
k〉ph
p
k. For r ≥ p,
Pn is a bounded operator from Hp to Hr. In addition, Pn is an n–dimensional orthogonal
projection on Hr, since, for u ∈ Hr, we have
Pn(u) =
n∑
k=1
〈u, hp
k〉ph
p
k =
n∑
k=1
〈u, hr
k〉r〈hr
k, h
p
k〉ph
p
k =
n∑
k=1
〈u, hr
k〉rhr
k.
Using condition [INV(Φ)], consider the coefficients PnL : [0, T ]×Ω → L(PnHr, PnHr)
and PnA : [0, T ] × Ω → L(PnHr, L(Rd, PnHr)), and a finite dimensional SDE
(2.6) Xn(t) = Pnφ+
∫ t
0
PnL(s)Xn(s) ds+
∫ t
0
PnA(s)Xn(s) dBs.
By [B(r,p)] and linearity, it is easy to see that the coefficients of this equation are
Lipschitz–continuous, so that, by the finite dimensional result (e.g., Theorem 3, Chapter
II, vol. 3, in Gikhman and Skorokhod [3]), there exists a strong solution Xn(t) in PnHr.
We verify that the coefficients PnL and PnA satisfy condition [M(r)] for u ∈ PnHr ⊂ Φ,
2〈PnL(s)u, u〉r + ‖PnA(s)u‖2
HS(r) ≤ 2〈L(s)u, u〉r + ‖Pn‖2‖A(s)u‖2
HS(r) ≤ θ‖u‖2
r,
due to the assumptions [INV(Φ)] and [M(r)], on L and A. Thus, by (2.3),
sup
n
sup
t≤T
E‖Xn(t)‖2
r ≤ CE‖φ‖2
r.
Hence, the sequenceXn is bounded in L2
(
Ω×[0, T ], Hr
)
, and we can select a subsequence,
denoted again by Xn, which converges weakly to an element X in L2
(
Ω × [0, T ], Hr
)
.
We can choose the limit X such that it has a progressively measurable modification
{Xt}0≤t≤T , since the limit in L2
(
Ω × [0, T ]) of the sequence {〈hr
i , Xn(t)〉r}∞n=1 viz.
〈hr
i , Xt〉r is progressively measurable for each i.
We now prove that the process {Xt}0≤t≤T satisfies SDE (2.2) in Hp by showing that,
in (2.6), we can replace Xn with X on the RHS and with PnX on the LHS.
Let η(s, ω) = η1(s)η2(ω)hp
i , where η1 and η2 are real–valued bounded and measurable.
Note that, for u ∈ Hp, 〈hp
i , u〉p = 〈hp
i , h
r
i 〉p〈hr
i , u〉r. So, using the weak convergence of
Xn to X in L2
(
Ω × [0, T ], Hr
)
, we obtain
E
∫ T
0
〈
η(s), Xn(s)
〉
p
ds→ E
∫ T
0
〈
η(s), Xs
〉
p
ds.
Note that, by condition [B(r,p)] and the boundedness of Xn in L2
(
Ω × [0, T ], Hr
)
, we
have
E
∣∣∣∣η2 ∫ s
0
〈
hp
i , L(u)Xn(u)
〉
p
du
∣∣∣∣ ≤ C and E
∣∣∣∣η2 ∫ s
0
〈hp
i , (A(u)Xn(u)) ej〉p du
∣∣∣∣ ≤ C,
where the constant C is independent of n and s.
By the weak convergence of Xn to X in L2
(
Ω × [0, T ], Hr
)
, it follows that
Eη2
∫ s
0
〈
hp
i , L(u)Xn(u)
〉
p
du = Eη2
∫ s
0
〈
L∗(u)hp
i , Xn(u)
〉
r
du
→ Eη2
∫ s
0
〈
L∗(u)hp
i , Xu
〉
r
du = Eη2
∫ s
0
〈
hp
i , L(u)Xu
〉
p
du.
LINEAR SDE’S IN THE DUAL OF A MULTI-HILBERTIAN SPACE 33
Now, by the Lebesgue DCT,
lim
n→∞E
∫ T
0
η1(s)η2(ω)
∫ s
0
〈
hp
i , PnL(u)Xn(u)
〉
p
du ds
= E
∫ T
0
η1(s)η2(ω)
∫ s
0
〈
hp
i , L(u)Xu
〉
p
du ds.
Let Aj(u) : Hr → Hp be defined by
Aj(u)hr
k = (A(u)hr
k) (ej).
Repeating the above arguments with the operator Aj replacing L proves that, for all i, j,
lim
n→∞Eη2
∫ T
0
η1(u)〈hp
i , (A(u)Xn(u))ej〉p du = Eη2
∫ T
0
η1(u)〈hp
i , (A(u)Xu)ej〉p du.
Thus, 〈hp
i , (A(u)Xn(u))ej〉p → 〈hp
i , (A(u)Xu)ej〉p weakly in L2(Ω × [0, T ]). By Doob’s
inequality, with a one–dimensional Brownian motion βt and a stochastically integrable
predictable process ξ(t), we have
E
∫ T
0
∣∣∣∣∫ s
0
ξ(u) dβu
∣∣∣∣2 ds ≤ TE
(
sup
0≤s≤T
∣∣∣∣∫ s
0
ξ(u) dβu
∣∣∣∣2
)
≤ TE
∫ T
0
|ξ(s)|2 ds,
which implies that the stochastic integral is a continuous linear operator from L2(Ω ×
[0, T ],P) to L2(Ω × [0, T ],FT ⊗ B[0, T ]) (here, P is the predictable σ–field, and B is
the Borel σ–field). By Theorem 15, [DS], Ch. V, §4, it is also continuous in the weak
topologies, so that
lim
n→∞E
∫ T
0
η1(s)η2(ω)
d∑
j=1
∫ s
0
〈
hp
i , (PnA(u)Xn(u))ej
〉
p
dBj
u ds
= E
∫ T
0
η1(s)η2(ω)
d∑
j=1
∫ s
0
〈
hp
i , (A(u)Xu)ej
〉
p
dBj
u ds.
To complete the proof, we multiply Eq. (2.6) by η(s) and integrate w.r.t. dP ×dt. Then,
by letting n→ ∞, we get, for a.e. (ω, t), dP × dt,
〈hp
i , Xt〉p = 〈hp
i , φ〉p +
∫ t
0
〈
hp
i , L(u)Xs
〉
p
ds+
d∑
j=1
∫ t
0
〈
hp
i , (A(u)Xs)ej
〉
p
dBj
s .
The process Xt has values in Hr, with X ∈ L2
(
Ω × [0, T ], Hr
)
⊂ L2
(
Ω × [0, T ], Hp
)
,
and satisfies Eq. (2.2) in Hp a.e. dP × dt. Thus, Xt is a strong Hr–valued solution of
Eq. (2.1) in Hp.
The continuity of {Xt}t≤T with respect to the initial condition follows from (2.4).
Example. The space S of smooth rapidly decreasing functions on Rd with the topol-
ogy given by L. Schwartz is nuclear. Let Sp be the completion of S with respect to the
Hilbertian norms ‖f‖2
p =
∑∞
|k|=0 (2|k| + d)2p 〈f, hk〉L2(Rd) , f, g ∈ S, where {hk}∞k=1 is
an ONB in L2
(
Rd, dx
)
given by Hermite functions. Then S′ =
⋃
p>0 S−p. Let {σij(t)}t≥0
and {bi(t)}t≥0 be bounded progressively measurable processes. Define, for ϕ ∈ S ′
,
L(t, ω)ϕ :=
1
2
d∑
i,j=1
(σσT )ij(t, ω) ∂2
ijϕ−
d∑
i=1
bi(t, ω) ∂iϕ
Ai(t, ω)ϕ :=
d∑
j=1
σji(t, ω) ∂jϕ,
34 L. GAWARECKI, V. MANDREKAR, AND B. RAJEEV
and let A(t, ω)ϕ ≡ (A1ϕ(t, ω), . . . Adϕ(t, ω)). Then A and L satisfy the conditions for
existence and uniqueness of the solution in Theorem 1 (for details, see Gawarecki et
al. [2]). Specifically, condition [M(r)] holds true for any r ∈ R, and condition [M(p,q)]
is satisfied for q ≤ p− 1. It is easy to verify using the recurrence properties of Hermite
polynomials that condition [B(r,p)] is valid for any p ≤ r − 1. Hence, setting r ≥ p+ 1,
and q ≤ p − 1, for any p ∈ R, and φ ∈ L2(Ω, Sr), Eq. (2.1) has a unique continuous
Sr–valued strong solution in Sp which is continuous in L2(Ω, C([0, T ], Sq)) with respect
to φn → φ in L2(Ω, Sp).
Consider a special case where Aϕ = (−∂1ϕ, . . . ,−∂dϕ) and Lϕ = 1
2
∑d
i=1 ∂
2
i ϕ. The
unique solution of Eq. (2.1) with the initial condition δx is δBt , where P (B0 = x) = 1.
This follows from the Itô formula in [8],
ρBtφ = ρB0φ−
d∑
i=1
∫ t
0
∂i (ρBsφ) dBi
s +
1
2
d∑
i=1
∫ t
0
∂2
i (ρBsφ) ds.
Here, for x ∈ Rd, ρx denotes the translation operator on Rd. If φ ∈ S′, then 〈f, ρxφ〉 :=
〈ρ−xf, φ〉 = 〈f(· + x), φ〉 for f ∈ S. For each t, ρBtφ denotes the S′–valued random
variable ω → ρBt(ω)φ. Then {ρBtφ}t≥0 is an S−p–valued stochastic process for some
p > 0, as shown in [8]. Taking φ = δ0 gives ρBtφ = δBt .
However, it is easy to verify that the coefficients A and L do not satisfy the coercivity
inequality in [6], and they violate the linear growth condition in [5].
Bibliography
1. N. Dunford, J.T. Schwarz, Linear Operators. Part I: General Theory, Interscience Publishers,
New York, 1958.
2. L. Gawarecki, V. Mandrekar, B. Rajeev, The monotonicity inequality for a pair of differential
operators, Infin. Dimens. Anal. Quantum Probab. Relat. Top. (submitted).
3. I.I. Gikhman, A.V. Skorokhod, The Theory of Stochastic Processes, Springer, New York, 1974.
4. K. Itô, Foundations of Stochastic Differential Equations in Infinite Dimensional Spaces,
CBMS–NSF 47 (1984).
5. G. Kallianpur, I. Mitoma, R.L. Wolpert, Diffusion equations in dual of nuclear spaces,, Stoch.
Stoch. Reports 29 (1990), 295–329.
6. N.V. Krylov, B.L. Rozovskii, Stochastic evolution equations, Itogi Nauki i Tekhniki, vol. 14,
Trans. by Plenum Publ. Corpor., 1981, pp. 1233–1277.
7. E. Pardoux, Stochastic Partial Differential Equations and Filtering of Diffusion Processes,
Stochastics 3 (1979), 127–167.
8. B. Rajeev, From Tanaka formula to Itô formula: distributions, tensor products and local times,,
Seminaire de Probabilites XXXV, LNM, vol. 1755, Springer, Berlin, 2001, pp. 371-389.
9. B. Rozovskii, Stochastic Evolution Systems: Linear Theory and Applications to Non-Linear
Filtering, Kluver Academic Publishers, Boston, 1983.
'��������� � %����������� !�������6 ���"�����
� #3�� 8� &���� �"��� (����� %�
$ 4�$� ������
E-mail : lgawarec@kettering.edu
'��������� � ���������� ��� ���0�0����
� %����6�� ����� ���"�����
� -��� /�����6�
%�� ������
E-mail : mandrekar@stt.msu.edu
����� %���� ����� ������ ����������� ���������� 9��6������ �����
E-mail : brajeev@isibang.ac.in
|
| id | nasplib_isofts_kiev_ua-123456789-4549 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0321-3900 |
| language | English |
| last_indexed | 2025-12-07T15:55:03Z |
| publishDate | 2008 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Gawarecki, L. Mandrekar, V. Rajeev, B. 2009-12-03T16:35:05Z 2009-12-03T16:35:05Z 2008 Linear stochastic differential equations in the dual of a multi-Hilbertian space / L. Gawarecki, V. Mandrekar, B. Rajeev // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 2. — С. 28–34. — Бібліогр.: 9 назв.— англ. 0321-3900 https://nasplib.isofts.kiev.ua/handle/123456789/4549 519.21 We prove the existence and uniqueness of strong solutions for linear stochastic differential equations in the space dual to a multi–Hilbertian space driven by a finite dimensional Brownian motion under relaxed assumptions on the coefficients. As an application, we consider equtions in S' with coefficients which are differential operators violating the typical growth and monotonicity conditions. en Інститут математики НАН України Linear stochastic differential equations in the dual of a multi-Hilbertian space Article published earlier |
| spellingShingle | Linear stochastic differential equations in the dual of a multi-Hilbertian space Gawarecki, L. Mandrekar, V. Rajeev, B. |
| title | Linear stochastic differential equations in the dual of a multi-Hilbertian space |
| title_full | Linear stochastic differential equations in the dual of a multi-Hilbertian space |
| title_fullStr | Linear stochastic differential equations in the dual of a multi-Hilbertian space |
| title_full_unstemmed | Linear stochastic differential equations in the dual of a multi-Hilbertian space |
| title_short | Linear stochastic differential equations in the dual of a multi-Hilbertian space |
| title_sort | linear stochastic differential equations in the dual of a multi-hilbertian space |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/4549 |
| work_keys_str_mv | AT gawareckil linearstochasticdifferentialequationsinthedualofamultihilbertianspace AT mandrekarv linearstochasticdifferentialequationsinthedualofamultihilbertianspace AT rajeevb linearstochasticdifferentialequationsinthedualofamultihilbertianspace |